Math 2.0

Measure and integration Definition

Standard probability space

Builds on Probability spaceLebesgue measureNull setComplete measure

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space whose measure is complete. A model space is built from a number a∈[0,1]a \in [0,1] and a finite or countable family of numbers pk>0p_k > 0 with a+∑kpk=1a + \sum_k p_k = 1. It is the disjoint union of [0,a][0, a] with points xkx_k. Lebesgue measurable subsets of [0,a][0,a] get their Lebesgue measure, and each point xkx_k gets mass pkp_k. Two complete probability spaces are isomorphic mod 00 if a null set can be removed from each so that the following holds. There is a bijection between the remaining sets that is measurable in both directions and preserves the measure. The space (Ω,ℱ,P)(\Omega, \mathcal{F}, P) is standard if it is isomorphic mod 00 to a model space.

Remarks

Example: the unit square [0,1]2[0,1]^2 with completed Lebesgue measure is standard with a=1a = 1 and no atoms. Remove the null set of points with a dyadic rational coordinate, and interleave the binary digits of the two coordinates. This gives a bijection mod 00 with [0,1][0,1] that sends dyadic squares to dyadic intervals of the same measure, so it preserves the measure. Non-example: let Z⊆[0,1]Z \subseteq [0,1] have Lebesgue inner measure 00 and outer measure 11, and let Ω=[0,1]∖Z\Omega = [0,1] \setminus Z with P(A∩Ω)=λ(A)P(A \cap \Omega) = \lambda(A) for Lebesgue measurable AA. This "perforated interval" is a complete probability space that is known not to be standard. Rokhlin's Lebesgue spaces are the same spaces, defined through a countable separating family of sets and a completeness condition.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptComplete measure, concept, no evidence levelComplete measureconceptEuclidean space, object, no evidence levelEuclidean spaceobjectNull set, concept, no evidence levelNull setconceptOuter measure, concept, no evidence levelOuter measureconceptProbability space, concept, no evidence levelProbability spaceconceptLebesgue measure, object, no evidence levelLebesgue measureobjectStandard probability space, definition, no evidence levelStandard probability …definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Complete measure concept
  3. Euclidean space object
  4. Null set concept
  5. Outer measure concept
  6. Probability space concept
  7. Lebesgue measure object
  8. Standard probability space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Standard_probability_space, Definition
About this page
Tag3H1YKSPH19
URIm2://3H1YKSPH19
Kinddefinition
WritingW1 stated
MSC28A05, 60A10, 28D05
DefinesStandard probability space
UsesProbability space, Lebesgue measure, Null set, Complete measure
Changed2 November 2026

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